$ k $th powers in a generalization of Piatetski-Shapiro sequences

Abstract

<abstract><p>The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c &gt; 1 $, and $ \beta $ are real numbers.</p> <p>The focus of the study is on solving equations of the form $ \left\lfloor \alpha n^c +\beta\right\rfloor = s m^k $, where $ m $ and $ n $ are positive integers, $ 1 \leq n \leq N $, and $ s $ is an integer. Bounds for the solutions are obtained for different values of the exponent $ k $, and an average bound is derived over $ k $-free numbers $ s $ in a given interval.</p></abstract>